Harmonic maps, Toda frames and extended Dynkin diagrams

Harmonic maps, Toda frames and extended Dynkindiagrams

Emma CarberryUniversity of Sydney

Katharine TurnerUniversity of Chicago

3rd of December, 2011

Emma Carberry and Katharine Turner Harmonic maps, Toda frames and extended Dynkin diagrams

Coxeter automorphism on GC/TC and conditions for it topreserve a real formde Sitter spheres S2n

1 and isotropic flag bundlesToda integrable system and relationship to cyclic primitivemaps from a surface into G/TSolution in terms of ODEs (finite type)Applications to superconformal tori in S2n

1

Applications to Willmore tori in S3.


Coxeter automorphism on GC/TC

Let GC be a simple complex Lie group and TC a Cartansubgroup.

The homogeneous space GC/TC is naturally a k -symmetricspace.

That is, we have an automorphism σ : GC → GC with σk = 1and

(GCσ )id ⊂ TC ⊂ GC

σ .

Recall that a non-zero α ∈ (tC)∗ is a root with root spaceGα ⊂ gC if

[H,Rα] = α(H)Rα ∀H ∈ t, Rα ∈ Gα.


Choose a set of simple roots, that is roots α1, . . . , αN suchthat every root can be written uniquely as

α =N∑

j=1

mjαj ,

where all mj ∈ Z+ or all mj ∈ Z−.

The height of α is h(α) =∑N

j=1 mj and the root of minimalheight is called the lowest root.

Let η1, . . . , ηN ∈ tC be the dual basis to α1, . . . , αN andσ : GC → GC be conjugation by

exp(2πik

N∑j=1

ηj) (Coxeter automorphism).

Then σ has order k , where k − 1 is the maximal height of a rootof gC.


Choose a set of simple roots, that is roots α1, . . . , αN suchthat every root can be written uniquely as

α =N∑

j=1

mjαj ,

where all mj ∈ Z+ or all mj ∈ Z−.

The height of α is h(α) =∑N

j=1 mj and the root of minimalheight is called the lowest root.

Let η1, . . . , ηN ∈ tC be the dual basis to α1, . . . , αN andσ : GC → GC be conjugation by

exp(2πik

N∑j=1

ηj) (Coxeter automorphism).

Then σ has order k , where k − 1 is the maximal height of a rootof gC.


Let G be a real simple Lie group with Cartan subgroup T andassume that the Coxeter automorphism preserves the real formG.

I will describe class of harmonic maps from the surface intoG/T which are given simply by solving ordinary differentialequations and give a relationship between these maps and theToda equations.

This will generalise work of Bolton, Pedit and Woodward for thecase when G is compact.


Example: SO(2n,1)

Let R2n,1 denote R2n+1 with the Minkowski inner product

x1y1 + x2y2 + · · ·+ x2ny2n − x2n+1y2n+1

Consider the de Sitter group SO(2n,1) of orientationpreserving isometries of R2n,1. Take as Cartan subgroup

T = diag (1,SO(2), . . . ,SO(2),SO(1,1)) .


Define ak ∈ t∗, k = 1, . . . ,n by

ak

(diag

0,(

0 a1−a1 0

), . . .

(0 anan 0

))= ak .

Take as simple roots of so(2n,1,C) the roots

α1 = i a1,

αk = i ak − i ak−1 for 1 < k < n andαn = an − i an−1.

The lowest root is then α0 = −an − i an−1, which is of height−2n + 1.


Then writing ηj for the dual basis of tC, conjugation by

Q = exp(πi

n

n∑j=1

ηj

)= diag

(1,R

(πn

),R(

2πn

), . . . ,R

( rπn

),−I2

)is an automorphism of order 2n.

It is not hard to prove directly in this case that the real formSO(2n,1) is preserved by the Coxeter automorphism.


Let 〈·, ·〉 denote the complex bilinear form

〈z,w〉 = z1w1 + z2w2 + · · ·+ z2nw2n − z2n+1w2n+1

on C2n+1.

A subspace V ⊂ C2n+1 is isotropic if 〈u, v〉 = 0 for all u, v ∈ V .

Geometrically,

SO(2n,1)/T = SO(2n,1)/(1×SO(2)×· · ·×SO(2)×SO(1,1))

is the full isotropic flag bundle

Fl(S2n1 ) = V1 ⊂ V2 ⊂ · · · ⊂ Vn−1 ⊂ TCS2n

1 | Vj is anisotropic sub-bundle of dimension j


We now give conditions under which a choice of

real form g of a simple complex Lie algebra gC,Cartan subalgebra tC and simple roots αj

yield a Coxeter automorphism σ = Adexp( 2πik

∑Nj=1 ηj )

whichpreserves the real Lie algebra g.


The condition for the Coxeter automorphism σ to preserve g isthat for the simple roots α1, . . . , αN we have

αj ∈ −α0, . . . ,−αN,

where α(X ) = α(X ) and α0 is the lowest root.

We will now use a Cartan involution to express this realitycondition in terms of the extended Dynkin diagram.


A Cartan involution for g is an involution Θ of gC such that

〈X ,Y 〉Θ = −〈X ,Θ(Y )〉

is positive definite on g, where 〈·, ·〉 denotes the Killing form.Alternatively, it is an involution for which

k⊕ im

is compact, where

k = +1-eigenspace of Θ

m = −1-eigenspace of Θ.

We may choose a Cartan involution which preserves the givenCartan subalgebra t


Proposition

Let g be a real simple Lie algebra, t a Cartan subalgebra and Θbe a Cartan involution preserving t. Choose simple rootsα1, . . . , αN and let σ = Adexp( 2πi

k∑N

j=1 ηj )be the corresponding

Coxeter automorphism of gC. Then the following are equivalent:

1 σ preserves the real form g,2 σ commutes with Θ,3 Θ defines an involution of the extended Dynkin diagram for

gC consisting of the usual Dynkin diagram augmented withthe lowest root α0.


For a Θ-stable Cartan subalgebra t,

t is maximallycompact

⇔ Θ defines a permutationof the Dynkin diagram for gC

and so when t is maximally compact (e.g. g is compact), thereal form g is preserved by any Coxeter automorphism definedby simple roots for t.

The more interesting case is when we have an involution of theextended Dynkin diagram which does not restrict to aninvolution of the Dynkin diagram (i.e. t is not maximallycompact).

Call these non-trivial involutions.


E8α0 α8 α7 α6 α5 α4 α3 α1

α2

DN . . .α1 α2

α0

αN−2

αN−1

αN

CN . . .α0 α1 α2 αN−1 αN

BN . . .α1 α2 αN−1 αN

α0

. . .

AN

α1 αN

αN−1α2

α0

E7 α7 α6 α5 α4 α3 α1 α0

α2

E6 α6 α5 α4 α3 α1

α2

α0

F4 α0 α1 α2 α3 α4

G2 α0 α1 α2

There are nontrivial involutions for all root systems exceptE8,F4 and G2.


Theorem

Every involution of the extended Dynkin diagram for a simplecomplex Lie algebra gC is induced by a Cartan involution of areal form of gC.

More precisely, let gC be a simple complex Lie algebra withCartan subalgebra tC and choose simple roots α1, . . . , αN forthe root system ∆(gC, tC). Given an involution π of theextended Dynkin diagram for ∆, there exists a real form g of gC

and a Cartan involution Θ of g preserving t = g ∩ tC such that Θinduces π and t is a real form of tC. The Coxeter automorphismσ determined by α1, . . . , αN preserves the real form g.


Primitive Maps and Loop Groups

The Coxeter automorphism σ : g→ g of order k induces aZk -grading

gC =k−1⊕j=0

gσj , [gσj , gσl ] ⊂ gσj+l ,

where gσj denotes the ej 2πik -eigenspace of σ.

We have the reductive splitting

g = t⊕ p

with

pC =k−1⊕j=1

gσj , tC = gσ0 ,

and if ϕ is a g-valued form we may decompose it asϕ = ϕt + ϕp.


A smooth map f of a surface into a symmetric space (G/K , σ)is harmonic if and only if for a smooth lift F : U → G off : U → G/K , the form ϕ = F−1dF has the property that foreach λ ∈ S1

ϕλ = λϕ′p + ϕk + λ−1ϕ′′p

satisfies the Maurer-Cartan equation

dϕλ +12

[ϕλ ∧ ϕλ] = 0.

Moreover given a family of flat connections as above, we canrecover a harmonic map f : U → G/K on any simply connectedU.


A smooth map f of a surface into a symmetric space (G/K , σ)is harmonic if and only if for a smooth lift F : U → G off : U → G/K , the form ϕ = F−1dF has the property that foreach λ ∈ S1



dϕλ +12

[ϕλ ∧ ϕλ] = 0.

Moreover given a family of flat connections as above, we canrecover a harmonic map f : U → G/K on any simply connectedU.


When k > 2, the condition that



dϕλ +12

[ϕλ ∧ ϕλ] = 0.

characterises (not merely harmonic but) primitive maps ψ of asurface into the k -symmetric space G/K .

ψ is primitive if for a smooth lift F : U → G of ψ : U → G/K ,ϕ′ = F−1∂F takes values in gσ0 ⊕ gσ1 . We say that F is aprimitive frame.

Primitive maps ψ are in particular harmonic.


For studying maps into G/T it is helpful to consider the twistedloop group

ΩσG = γ : S1 → G : γ(e2πik λ) = σ(γ(λ))

and corresponding twisted loop algebra Ωσg. The (possiblydoubly infinite) Laurent expansion

ξ(λ) =∑

j

ξjλj , ξj ∈ gσj ⊂ gC, Φ−j = Φj

allows us to filtrate ΩσgC by finite-dimensional subspaces

Ωσd = ξ ∈ Ωg | ξj = 0 whenever |j | > d.


Suppose ξ : R2 → Ωσd satisfies the Lax equation

∂ξ

∂z= [ξ, λξd +

12ξd−1].

Then

ϕλ(z) =(λξd (z) +

12ξd−1(z)

)dz +

(λ−1ξ−d (z) +

12ξd−1(z)

)dz

satisfies the Maurer-Cartan equation and so defines a primitivemap f : R2 → G/T .

Maps f obtained in this simple way are said to be of finite type.


Suppose ξ : R2 → Ωσd satisfies the Lax equation

∂ξ

∂z= [ξ, λξd +

12ξd−1].

Then


12ξd−1(z)

)dz +

(λ−1ξ−d (z) +

12ξd−1(z)

)dz


Maps f obtained in this simple way are said to be of finite type.


The equation

12

(X (ξ)− iY (ξ)) =(λξd +

12ξd−1

)defines vector fields X ,Y on Ωσ

d .

Assume the vector fields X ,Y are complete (e.g. G iscompact). The vector fields X ,Y commute and so define anaction

(x , y) · ξ(λ) = X x Y y (ξ(λ))

of R2 on Ωd . Define ξ(z, λ) := (x , y) · ξ0(λ) for any initialξ0(λ) ∈ Ωd , where z = x + iy . Then


12ξd−1(z)

)dz +

(λ−1ξ−d (z) +

12ξd−1(z)

)dz



For the Coxeter automorphism on G/T , gσ0 = t and gσ1 is thesum of the simple and lowest root spaces.

We say that a primitive map ψ / frame F is in addition cyclic ifthe image of F−1∂F contains a cyclic element.

An element of gσr0 ⊕ gσr

1 is cyclic if its projection to each of theroot spaces Gα1 , . . . ,Gαn ,Gα0 is non-zero.

I will now describe the relationship between cyclic primitivemaps into G/T and the Toda equations.


For the Coxeter automorphism on G/T , gσ0 = t and gσ1 is thesum of the simple and lowest root spaces.

We say that a primitive map ψ / frame F is in addition cyclic ifthe image of F−1∂F contains a cyclic element.

An element of gσr0 ⊕ gσr

1 is cyclic if its projection to each of theroot spaces Gα1 , . . . ,Gαn ,Gα0 is non-zero.

I will now describe the relationship between cyclic primitivemaps into G/T and the Toda equations.


Toda equationThe classical 1-dimensional affine Toda integrable systemdescribes the motion of finitely many particles of equal massarranged in a circle, joined by “exponential springs".

0

1 n

2 n − 1

. . . . ..

md2xj

dt2 = e(xj−1−xj ) − e(xj−xj+1).


We may generalise this to any simple Lie algebra as

2d2Ω

dt2 =n∑

j=0

mje2αj (Ω)[Rαj ,R−αj ]

or for a 2-dimensional domain

2Ωzz =N∑

j=0

mje2αj (Ω)[Rαj ,R−αj ] (1)

where Ω : C→ it is a smooth map, mj ∈ R+ satisfies mπ(j) = mj

and Rαj are root vectors satisfying Rαj = R−απ(j) .


To recover the classical Toda equation:

1 Take the standard simple roots for su(n + 1).2 Set m0 = 1 and let

α0 = −N∑

j=1

mjαj

be the expression for the lowest root α0.3 Choose root vectors Rαj so that [Rαj ,R−αj ] is the dual of αj

with respect to the Killing form.

Notice that the extended Dynkin diagram for su(n + 1) looks like

. . .

α1 αN

αN−1α2

α0


Given a cyclic element W =∑N

j=0 rjRαj of gσ1 , we say that a liftF : C→ G of ψ : C→ G/T is a Toda frame with respect to W ifthere exists a smooth map Ω : C→ it such that

F−1Fz = Ωz + Adexp ΩW .

We call Ω an affine Toda field with respect to W .

Lemma

The affine Toda field equation (1) is the integrability conditionfor the existence of a Toda frame with respect to W.

Here W =∑N

j=0 rjRαj is a cyclic element of gσ1 such thatmπ(j) = mj and Rαj = R−απ(j) and we take mj = rj rj forj = 0, . . . ,N.


Toda frame and cyclic primitive

Theorem

A map ψ : C→ G/T possesses a Toda frame if and only if ithas a cyclic primitive frame F for which c0

∏Nj=1 cmj

j is constant,where

F−1Fz |gσ1 =N∑

j=0

cjRαj .

The Toda frame is then cyclic primitive with respect to anyW =

∑Nj=0 rjRαj for which

r0

N∏j=1

rmjj = c0

N∏j=1

cmjj .


Finite-type

Theorem

Let G be a simple real Lie group, T a Cartan subgroup andassume that the Coxeter automorphism preserves G. Supposeψ : C/Λ→ G/T has a Toda frame F : C/Λ→ G. Then ψ is offinite type.


Harmonic maps into S2n1

The isotropy order of a harmonic map f of a surface into S2n1 is

the maximal integer r ≥ 0 such that the derivatives∂zF , ∂2

z F , . . . , ∂rz f span an isotropic subspace at each point.

If f has the maximal isotropy order r = n we say it is isotropic.

Isotropic surfaces in S2n1 include all harmonic maps of S2, and

can be expressed holomorphically in terms of aWeierstrass-type representation (Bryant 84, Ejiri 88)

Harmonic maps f : M2 → S2n1 with the penultimate isotropy

order r = n − 1 are said to be superconformal.


Applying Gram-Schmidt, we define the harmonic sequencef0, f1, . . . , fr of a non-constant harmonic map f : M2 → S2n

1 by

f0 = f , fj+1 = ∂z fj −〈∂z fj , fj〉‖fj‖2

fj wherever ‖fj‖2 6= 0

and extend by continuity wherever fj = 0. Then

∂z fj+1 = −‖fj+1‖2

‖fj‖2fj for 0 ≤ j ≤ r

〈fj , fk 〉 = 0 unless j = k

and the zeros of the fj are isolated whenever fj does not vanishidentically (Hulett 05).


Applying Gram-Schmidt, we define the harmonic sequencef0, f1, . . . , fr of a non-constant harmonic map f : M2 → S2n

1 by

f0 = f , fj+1 = ∂z fj −〈∂z fj , fj〉‖fj‖2

fj wherever ‖fj‖2 6= 0

and extend by continuity wherever fj = 0. Then

∂z fj+1 = −‖fj+1‖2

‖fj‖2fj for 0 ≤ j ≤ r

〈fj , fk 〉 = 0 unless j = k

and the zeros of the fj are isolated whenever fj does not vanishidentically (Hulett 05).


Theorem

A harmonic map f : C→ S2n1 has a cyclic primitive lift

ψ : C→ Fl(S2n1 ) if and only if it is superconformal and the

entries f1, . . . , fn−1 of its harmonic sequence are definedeverywhere.

We have for each 1 ≤ j ≤ r

fj = 2j−1c1 . . . cjF (e2j + ie2j+1) for each 1 ≤ j ≤ n − 1

where the cj are root vector coefficients with respect toparticular choices of the root vectors appearing in gσr

1 .

Corollary

Let f : C/Λ→ S2n1 be a superconformal harmonic map with

globally defined harmonic sequence f1, . . . , fn. Then f has alift ψ : C/Λ→ SO(2n,1)/T of finite type.


Theorem

A harmonic map f : C→ S2n1 has a cyclic primitive lift

ψ : C→ Fl(S2n1 ) if and only if it is superconformal and the

entries f1, . . . , fn−1 of its harmonic sequence are definedeverywhere.

We have for each 1 ≤ j ≤ r

fj = 2j−1c1 . . . cjF (e2j + ie2j+1) for each 1 ≤ j ≤ n − 1

where the cj are root vector coefficients with respect toparticular choices of the root vectors appearing in gσr

1 .

Corollary

Let f : C/Λ→ S2n1 be a superconformal harmonic map with

globally defined harmonic sequence f1, . . . , fn. Then f has alift ψ : C/Λ→ SO(2n,1)/T of finite type.


An immersed surface φ : M2 → R3 is Willmore if it is critical forthe Willmore functional

W =

∫M2

H2 dA,

where H denotes the mean curvature of φ and dA the areaform.

Due to Gauss-Bonnet, it is equivalent to seek critical surfacesfor ∫

M2(H2 − K ) dA,=

∫M2

(k2 − k1)2 dA

where K is the Gauss curvature and k1, k2 are the principalcurvatures.

This latter functional is clearly conformally invariant and so weinstead consider immersions into S3.


An immersed surface φ : M2 → R3 is Willmore if it is critical forthe Willmore functional

W =

∫M2

H2 dA,

where H denotes the mean curvature of φ and dA the areaform.

Due to Gauss-Bonnet, it is equivalent to seek critical surfacesfor ∫

M2(H2 − K ) dA,=

∫M2

(k2 − k1)2 dA

where K is the Gauss curvature and k1, k2 are the principalcurvatures.

This latter functional is clearly conformally invariant and so weinstead consider immersions into S3.


It is not hard to show thatW(M2) ≥ 4π, with equality if and onlyif M2 is a (round) sphere.

The Willmore conjecture proposes thatW(C/Λ) ≥ 2π2 for anyimmersed torus with equality if and only if the torus isconformally equivalent to


It is not hard to show thatW(M2) ≥ 4π, with equality if and onlyif M2 is a (round) sphere.

The Willmore conjecture proposes thatW(C/Λ) ≥ 2π2 for anyimmersed torus with equality if and only if the torus isconformally equivalent to


The conformal Gauss map of an immersion φ : M2 → S3

associates to each point on the surface M2 its central sphere,that is the oriented 2-sphere in S3 with the same normal vectorand mean curvature.


A 2-sphere in S3 is the intersection of S3 and a hyperplane inR4;

S3 ∩ x1, x2, x3, x4 : a1x1 + a2x2 + a3x3 + a4x4 − b = 0.

For this hyperplane to intersect with S3 at more than one pointrequires a2

1 + a22 + a2

3 + a24 − b2 > 0 and hence we can scale

(a1,a2,a3,a4,b) so that a21 + a2

2 + a23 + a2

4 − b2 = 1.

De Sitter space S2n1 is the unit sphere in R2n+1 with respect to

the Minkowski metric

x1y1 + x2y2 + · · ·+ x2ny2n − x2n+1y2n+1.

Thus each 2-sphere in S3 can be identified with two antipodalpoints ±(a1,a2,a3,a4,b) ∈ S4

1 .

Choosing an orientation for the 2-sphere gives a well-definedelement of S4

1 .


A 2-sphere in S3 is the intersection of S3 and a hyperplane inR4;

S3 ∩ x1, x2, x3, x4 : a1x1 + a2x2 + a3x3 + a4x4 − b = 0.

For this hyperplane to intersect with S3 at more than one pointrequires a2

1 + a22 + a2

3 + a24 − b2 > 0 and hence we can scale

(a1,a2,a3,a4,b) so that a21 + a2

2 + a23 + a2

4 − b2 = 1.

De Sitter space S2n1 is the unit sphere in R2n+1 with respect to

the Minkowski metric

x1y1 + x2y2 + · · ·+ x2ny2n − x2n+1y2n+1.

Thus each 2-sphere in S3 can be identified with two antipodalpoints ±(a1,a2,a3,a4,b) ∈ S4

1 .

Choosing an orientation for the 2-sphere gives a well-definedelement of S4

1 .


Hence we see that the space of oriented 2-spheres in S3 isnaturally identified with S4

1 .

The conformal Gauss map f : M2 → S41 is given explicitly by

f (z) = H(z) · Φ(z) + N(z)

where Φ(z) = (φ(z),1), N = (n,0).


The conformal Gauss map f is weakly conformal and animmersion away from the umbilic points of φ.

The area form on M2 induced by f is given by (H2 − K )dA

Thus φ : M2 → S3 is a Willmore immersion without umbilicpoints if and only if f : M2 → S4

1 is a minimal immersion, orequivalently is conformal and harmonic.


A minimal immersion f : M2 → S41 can only have isotropy order

r = 1 (superconformal) or r = 2 (isotropic).

Recall that the second fundamental form of f isII(X ,Y ) = (∇X Y )⊥, where ⊥ denotes projection to theorthogonal complement of TM2 in TS4

1 .

The curvature ellipse of f at p ∈ M2 is the image of the unitcircle in TpM2 under the second fundamental form.

It is a circle precisely when 〈fzz(p), fzz(p)〉 = 0. This quantity isholomorphic, hence constant when M2 is compact.

The curvature ellipse of f is thus a circle precisely when f isisotropic. All isotropic f have been constructed by Bryant usingholomorphic data.


A minimal immersion f : M2 → S41 can only have isotropy order

r = 1 (superconformal) or r = 2 (isotropic).

Recall that the second fundamental form of f isII(X ,Y ) = (∇X Y )⊥, where ⊥ denotes projection to theorthogonal complement of TM2 in TS4

1 .

The curvature ellipse of f at p ∈ M2 is the image of the unitcircle in TpM2 under the second fundamental form.

It is a circle precisely when 〈fzz(p), fzz(p)〉 = 0. This quantity isholomorphic, hence constant when M2 is compact.

The curvature ellipse of f is thus a circle precisely when f isisotropic. All isotropic f have been constructed by Bryant usingholomorphic data.


We have seen that the first ellipse of curvature being anon-circular ellipse corresponds to f being superconformal.

For superconformal f : M2 → S41 the cyclic primitive frame F

constructed previously consists of

F = (f , fx , fy , v ,w)

where the last two columns of F are determined by theprincipal directions of the curvature ellipse.

Corollary

A Willmore immersion φ : T 2 → S3 without umbilic points maybe constructed either

1 from holomorphic Weierstrass data2 by integrating a pair of commuting vector fields on a

finite-dimensional space


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Harmonic maps, Toda frames and extended Dynkin diagrams

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